Polynomial differential equations over the quaternions

Abstract

We study some fundamental properties of autonomous polynomial differential equations over the quaternions H. After showing that every polynomial differential equation in R4 can be rewritten as a differential equation over H, we specialize the investigation to quaternionic differential equations with coefficients in C⊆H, which retain some properties of polynomial differential equations in C. We provide a detailed characterization of this class and discuss their stationary points and periodic orbits, including those at infinity. As an application of our theory we investigate the dynamics of generic quadratic systems of this type and of a special class of quaternionic Riccati differential equations with complex coefficients. Here we obtain a new type of Liouville–Arnold integrable system.